On some discrete qualitative properties of implicit finite difference solutions of nonlinear parabolic problems

Abstract

In this paper we investigate two special qualitative properties of the finite difference solutions of one-dimensional nonlinear parabolic initial boundary value problems. The first property says that the number of the so-called L-level points, or specially the number of the zeros, of the solution must be non-increasing in time. The second property requires a similar property for the number of the local maximizers and minimizers. First we recall a theorem that guarantees the above properties for the solution of a special second order nonlinear parabolic problem. Then we generate the numerical solution with the implicit Euler finite difference method and show that the obtained numerical solution satisfies the discrete versions of the above properties without any requirements on the mesh parameters. We close the paper with some numerical tests.